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Abstract

We investigate the impact of short optical feedback on a two-state quantum dot laser. A region in the feedback parameter space is identified, where the laser emission periodically alternates between oscillation bursts from the quantum dot ground and excited state, i.e. two-color anti-phase oscillation bursts. We compare these results to the low-frequency fluctuations and regular pulse packages of single-color semiconductor lasers and show via an in-depth bifurcation analysis, that the two-color oscillation bursts originate from a torus-bifurcation of a two-state periodic orbit. A cascade of further period-doubling bifurcations produces chaotic dynamics of the burst envelope. Our findings showcase the rich dynamics and complexity, which can be generated via the interaction of electronic and photonic time scales in quantum dot lasers with optical feedback.

Figures (7)

Fig. 1. (a) Sketch of the setup. A frequency selective mirror only couples back light at the frequency of the GS emission. (b) Sketch of the band structure across one quantum dot (QD). Charge carrier variables are denoted in green and their coupling via scattering processes is indicated by dark green arrows. Recombination of GS and ES charge carriers is represented by red and blue arrows respectively.

Fig. 2. (a) Input-output characteristics of the solitary two-state laser with the pump current normalized to the threshold current. The GS emission (orange) exhibits a rollover after the onset of ES emission (purple) and full quenching at higher pump currents. Operating point at $J = 1.4J_\textrm {GS}^\textrm {th}$ with $P_0 = 20.17\,\textrm {mW}$. (b) Feedback induced dynamics in the feedback strength, feedback phase plane at $\tau = 150\textrm {ps}$. The steady output-power at the GS frequency is color coded in orange. Blue and green colors indicate period one and higher order oscillations. Hatches denote two-state lasing. Enclosed by a two-state lasing region, the emission completely switches from the GS to the ES. The purple line represents the parameter range shown in the bifurcation scan Fig. 3. (c) Time series of the GS (orange) and ES (purple) emission power exemplifying the complex oscillations in the green region in (b) at $C=0.849,K=0.127$.

Fig. 5. Time traces and phase-space projections within the oscillation bursts regime for $K \in \{0.1210, 0.1220, 0.1250, 0.1295, 0.1305\}$. Left column (a1)-(e1): time traces of the GS (orange and green) and ES power (purple); prange and green lines correspond to the same trajectory, which is either close to the first GS ECM or to the ghost of the two-state ECM. Middle column (a2)-(e2): trajectory in a phase-space projection onto the GS inversion and instantaneous frequency shift $\langle \delta \nu \rangle _{\tau }$. Open orange circles represent the unstable first GS ECM, green circles the two-state ECM, green diamonds the transcritical bifurcation of the two-state ECM, blue stars the saddle-node bifurcation of the second GS ECM and the blue circle the stable second GS ECM as introduced in Fig. 3. Right column (a3)-(e3): trajectory projected onto the GS power, ES power, GS inversion space. Orange lines represent the Poincaré map created by the intersection of the trajectories with surfaces of constant ES powers (gray surfaces).

Fig. 6. Zoom of the bifurcation diagram in Fig. 3(a) and Fig. 4(b) showing local maxima of the GS power (a) and power spectra of the GS emission (b). A period-doubling cascade leads to chaotic motion of the oscillation bursts slow envelope. Vertical gray dashed lines indicate the feedback parameters used in Figs. 5(d)–5(e) and Fig. 7 respectively.

Fig. 7. Low frequency chaos produced by the dynamics of the bursting oscillations envelope at $K=0.12985$. Time series of the GS and ES power at two different time scales (a), power spectrum of the GS at two different frequency scales (b), phase-space projection onto the GS inversion and instantaneous frequency shift $\langle \delta \nu \rangle _{\tau }$ (c) and Poincaré map of the GS power $P^\textrm {GS}$ and inversion $\rho _\textrm {inv}$ produced by the intersection with the surface of constant ES power (d).

Table 2.

Fit parameters for charge-carrier scattering processes, extracted from microscopic calculations for a GS confinement energy of 64(35) meV and a GS-ES separation of 50(20) meV for electrons (holes), and T=300 K.

b:

electrons

holes

m:

GS

ES

GS

ES

Am,b (10−11cm2ns−1)

18.5

48.3

0.525

1.07

Bm,b (1011cm−2)

1.9

0.48

5.3

1.8

Cb (ns−1)

1014

2272

Db (1011cm−2)

1.4

2.3

D2D (1011cm−2eV−1)

180

1880

Tables (2)

Table 1.

Parameters used for numerical calculations unless noted otherwise.

Symbol

Value

Symbol

Value

gGS

230ns−1

WGS

0.44ns−1

gES

460ns−1

WES

0.55ns−1

κGS

80ns−1

β

2.2×10−3

κES

10ns−1

Rlossw

0.54×1011cm2ns−1

ZQD

3.0×107

ηGS2

101.1V2cm−2

NQD

1011cm−2

ηES2

108.5V2cm−2

fa,fi

0.5,0.5

δωES

125ns−1

νGS,νES

1,2

δωQWe

11.3×10−11cm2ns−1

τ

150ps

δωQWh

5.5×10−11cm2ns−1

Vmode

2.88×10−5mm−3

ϵb

14.2ϵ0

Table 2.

Fit parameters for charge-carrier scattering processes, extracted from microscopic calculations for a GS confinement energy of 64(35) meV and a GS-ES separation of 50(20) meV for electrons (holes), and T=300 K.